Lady Romana is quoted as saying a TARDIS weights 5x10^6 kilograms, but I've also read that she was mistaken in this or that she was referring to the inside.

The Doctor's TARDIS has been moved a number of times and has landed on soft ground and not sunk in at all. This indicates the full weight of the interior is not felt from the exterior. But in The Parting of the Ways, Rose, Jackie, and Mickey use a large truck with chains to free up part of the interior of the TARDIS, and when they pull with the truck, the TARDIS doesn't just get pulled along on the ground, so, at least when the door is open, there's some connection between the weight inside and the outside shell.

I've searched for this and Romana's answer is the only one I find, but if that were true and it weight that much on the outside, people could not pick it up and move it.

In Day of the Doctor UNIT was able to lift the TARDIS with a helicopter to transport it to the museum, i'd say if you identify that helicopter's specs (cause i don't think UNIT was using an alien enhanced helicopter) you can get the max weight the TARDIS could weigh on the outside, it would be slightly heavier than a normal box of that make since the panels on the front door hide a phone, who knows what other things they hide
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Memor-XJun 12 '14 at 0:01

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The outside cannot be the same weight as the inside, as the inside is often canonically infinite in size. (Only Doctor Who could have "often canonically" without it being wrong.)
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PhoshiJun 12 '14 at 9:17

6 Answers
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I've never seen a in-universe explanation (Possibly because that's something they have avoided defining, lest it cause complications later), but observing it's behavior, I'd guess the following:

The external shell has a weight consistent with what the chameleon circuit has modeled it upon. For example, the Doctor's Tardis is set up as a police box, and, being made primarily of wood, isn't that heavy. Other disguised Tardis examples, however, weren't so light -- The Master's Tardis, disguised as the Melkur, should have weighed about the same as a stone statue of it's size. That being said, I've never seen anything to really confirm this, it's just supposition.

As to when Mickey, Rose and Jackie were pulling on the console, the cable they connected was crossing the threshold into the main Tardis; they were pulling against the actual full weight of the Tardis, not just it's exterior shell. (Goes back to the whole Dimensionally transcendental thing; just as the exterior shape doesn't pass back physical requirements to the interior shape, so does force exerted on the inside not translate to force on the outside) Had they simply hooked the cable around the Tardis, it most likely would have moved quite easily. One way to think of it is that the door of the Tardis is an opening to a Pocket Universe; it doesn't CONTAIN it, so much as provide a portal to it.

Also remember that it can be moved without too much difficulty. It seems like a forklift is all that it takes to move it, and it occasionally seen being transported on the back of a truck.
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erdiedeDec 18 '11 at 17:17

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Yep; and either in the Tom Baker or Peter Davidson era, I seem to remember a group people picking it up and carrying it.
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K-H-WDec 18 '11 at 17:18

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Although the exterior does in some way contain it, as seen when the Titanic crashed into it. The bow of the Titanic protuded into the interior of the TARDIS without passing through the door.
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XantecDec 18 '11 at 19:12

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Yeah; that kind of bothered me, since it's not consistent with the rest of the series.. As I recall, he did mention forgetting to turn on the Force Field; it's possible (and perhaps logical) that the control rooms occupy a space at the perimeter of the interior structure -- if so, it may be 'mapped' to the outside of the Tardis to allow for other exits / ports. (Like the Melkur's eyes in Master's Tardis.') If so, you could break into any part of the exterior and enter the control room, with the force field off.
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K-H-WDec 18 '11 at 19:31

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@KeithHWeston: I can remember two instances, both in the Davidson era: Black Orchid, where the police of 1925 cart the Tardis off; and Frontios, where the Tractators drag it underground (that might not count though, as they have power over gravity so might have been able to move it no matter how much it weighed).
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Daniel RosemanDec 19 '11 at 9:30

In the Hartnell era a group of egyptian slaves are easily able to drag the TARDIS into a tomb.

And then again in the Troughton era during the serial The Faceless Ones the airport security are able to load the TARDIS onto the back of a truck and carry it off. I like the explanation that the exterior weighs as much as the object the chameleon circuit has chosen to replicate.

It seems hard to argue with this. The "chameleon" weight is, simply, just the same as the other "chameleon" properties. So, it looks like an XYZ, it appears to be made of XYZ (perhaps wood, stone, metal, whatever), the color is XYZ (blue, black, whatever), and indeed the weight would simply match these things. Further, it has indeed been moved, handled exactly in this way, many times. Seems pretty definitive!
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Joe BlowOct 23 '14 at 8:03

And here's a small pont: consider the chameleon "door". Note that it works and feels exactly like a normal door -- so, we know that the door weighs the usual, say, 20kg or so that a "normal" door weighs. It makes sense the other chameleon parts (so, the door hinges, the door handle, the door frame, the other three walls, etc) are all, also, "normal-like", ie they would just be "normal" weight and so on, just as we know the door is "normal".
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Joe BlowOct 23 '14 at 8:05

Coming a bit late to the discussion, I just want to point out that in the episodes Planet of Fire (season 21), The Master's TARDIS turned into a "pillar of stone" which fell over during one of the earthquakes. He later enlisted the help of the locals to put it back upright which took four people and another with a rope (and I assume a pulley).

I therefore think that a TARDIS weighs as much as the object the chameleon-circuit makes it turn into would weigh.

This doesn't seem to improve upon the accepted answer from two and a half years ago which says, "The external shell has a weight consistent with what the chameleon circuit has modeled it upon." It also mentioned The Master's TARDIS. I suggest checking out the Tour to get a better idea of how to ask and answer questions. We're not a typical discussion forum. I'd also suggest reading the answers in place before contributing an answer, to avoid duplicating an existing answer. Don't be discouraged, we were all new here at some point.
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Meat TrademarkJun 11 '14 at 15:36

The interior of the TARDIS is another dimension, whose entrance is the police telephone box, but the relation stop there. It is not contained there, and also, the TARDIS is equiped with gravity manipulators (that is how she fly trough normal space), so it could weight what the Doctor/she likes. I have to point, that the Exterior Shell isn't really wood, its a plasmic shell that LOOKS like wood, making impossible to guess the weight by the materials. Also, the Doctor could lock the TARDIS position so it wouldn't move; not allowing Rose, Jackie and Mickey to move her when they tried to open the console. Note that the part of the TARDIS dimension that is the console room is fully integrated with the External Shell, like we seen in the Voyage of the Dammed, when the Titanic broke into the console. Whith these I mean that the gravity manipulator have to be online all the time, allowing the TARDIS to weight what she liked or to the locked, because otherwise it would weight more or less as the Control Room.

In the Episode "Flatline," the twelfth doctor says that he has altered the relative gravity of the Tardis. He says this is always happening - and that the earth wouldn't be able to support its real weight if he didn't. By altering the "gravity" of the Tardis, Clara is thence able to place the now miniature Tardis (looking suspiciously like a plastic cookie jar) in her purse.

As such, the correct answer is "Whatever the Doctor wants it to weigh."